Question

A sample of 81 observations has a mean 23 and standard deviation                                

 3.6, Test the Null Hypothesis H₀ : μ = 25 against the alternative

 H₁:  μ < 25, assuming α = .01

Answer 1

$n=81 \\ \bar{x} = 23 \\ s = 3.6 \\ H_0: \mu = 25 \\ H_1: \mu < 25$

Test-statistic

$t= \frac{\bar{x} - \mu}{s / \sqrt{n}} \\ t = \frac{23-25}{3.6 / \sqrt{81} } \\ = \frac{-2}{0.4333} \\ = -1.566 \\ α=0.01 \\ df=n-1=81-1=80$

One-tailed test

$t_{crit} = 2.373$

Reject the null hypothesis if $t ≤ -t_{crit}$

$t= -1.566 > t_{crit} = -2.373$

We accept the null hypothesis at 0.01 level of significance.